A Faddeev Calculation for Pentaquark ${\mathrm{\Theta }}^{+}$ in Diquark Picture with Nambu-Jona-Lasinio Type Interaction

A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark ${\mathrm{\Theta }}^{+}$ in the diquark picture of Jaffe and Wilczek in which ${\mathrm{\Theta }}^{+}$ is a diquark-diquark-$\bar{s}$ three-body system. Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two-body scatterings of $\bar{s}D$ and $DD$. With the use of coupling constants determined from the meson sector, we find that $\bar{s}D$ interaction is attractive while $DD$ interaction is repulsive, and there is no bound ${\frac{1}{2}}^{+}$ pentaquark state. A bound pentaquark ${\mathrm{\Theta }}^{+}$ can only be obtained with unphysically strong vector mesonic coupling constants.